Fractional Navier-Stokes-type model

deSolve package for ODE solutions

Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:18 PM (14 minutes ago)

to David, bcc: aimee

Sir,

A simple approach to fitting fractional differential equations in code that we can use to fit a fractional differential system on a graph. The R package deSolve does not handle fractional equations so I will do a code search since I’d rather not implement the code myself as many papers I’ve looked at give algorithms but perhaps this is a task that I can take on. The attached paper shows an example of numerical solution of a fractional differential equation. I did not succeed in finding code in an R or Octave package for this yet.

Possible task: implement direct modeling for solving and simulating deterministic fractional differential equations. This is a concrete task that will allow us to explore the exact models for volatility dynamics we hope.

Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:33 PM (0 minutes ago)

to David

An example makes the ‘variation iteration method’ pretty clear. One needs at each step the ability to be able to calculate the appropriate fractional derivatives. Now fractional derivatives and their generalizations can be coded directly from a numerical integration whether we take one of the several variations for the definition of fractional derivative defined by

(d/dt)^alpha u(t) = const \int_0^t (t-s)^{alpha-1} u(s)ds

in R for example:

fracdiff<-function( u, alpha, t0){

# t0=index

c<-1/gamma(alpha)

v<-0

for (i in 1:t0){

v<-v+pow(abs(float(i)-t0)),alpha-1)*u[i]

}

v
}

Now the iterative method is simply updating solutions with updates which involve this fractional derivative and other terms in the equation. These have good convergence properties.

Now we’re interested in actually using data to estimate the fractional differential equation that will be the first law of volatility say, so we fix the Hurst parameter (giving us exact value for alpha) and we estimate F(u) using the linear model we’ve begun with. So then we’re almost to a law for volatility without any idea of how it will fit.

Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:18 PM (14 minutes ago)

to David, bcc: aimee

Sir,

A simple approach to fitting fractional differential equations in code that we can use to fit a fractional differential system on a graph. The R package deSolve does not handle fractional equations so I will do a code search since I’d rather not implement the code myself as many papers I’ve looked at give algorithms but perhaps this is a task that I can take on. The attached paper shows an example of numerical solution of a fractional differential equation. I did not succeed in finding code in an R or Octave package for this yet.

Possible task: implement direct modeling for solving and simulating deterministic fractional differential equations. This is a concrete task that will allow us to explore the exact models for volatility dynamics we hope.

Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:33 PM (0 minutes ago)

to David

An example makes the ‘variation iteration method’ pretty clear. One needs at each step the ability to be able to calculate the appropriate fractional derivatives. Now fractional derivatives and their generalizations can be coded directly from a numerical integration whether we take one of the several variations for the definition of fractional derivative defined by

(d/dt)^alpha u(t) = const \int_0^t (t-s)^{alpha-1} u(s)ds

in R for example:

fracdiff<-function( u, alpha, t0){

# t0=index

c<-1/gamma(alpha)

v<-0

for (i in 1:t0){

v<-v+pow(abs(float(i)-t0)),alpha-1)*u[i]

}

v
}

Now the iterative method is simply updating solutions with updates which involve this fractional derivative and other terms in the equation. These have good convergence properties.

Now we’re interested in actually using data to estimate the fractional differential equation that will be the first law of volatility say, so we fix the Hurst parameter (giving us exact value for alpha) and we estimate F(u) using the linear model we’ve begun with. So then we’re almost to a law for volatility without any idea of how it will fit.